Bijections from Weighted Dyck Paths to Schröder Paths

Kim and Drake used generating functions to prove that the number of
2-distant noncrossing matchings, which are in bijection with little
Schröder paths, is the same as the weight of Dyck paths in which downsteps
from even height have weight 2. This work presents bijections from
those Dyck paths to little Schröder paths, and from a similar set of Dyck
paths to big Schröder paths. We show the effect of these bijections on the
corresponding matchings, find generating functions for two new classes
of lattice paths, and demonstrate a relationship with $231$-avoiding
permutations.